Triplet and Higher-Order Distribution Functions

By a straightforward generalization of the arguments given above, we can compute any higher-order distribution function. For instance, for a consecutive triplet of units, we have 

This should be compared with (4.3.47). This result can be expressed in terms of the singlet and pair distribution functions as follows  

This factoring of the triplet distribution function into pair and single distribution functions is a characteristic feature of the one-dimensional model. In terms of conditional probability, the same result can be stated as follows  

In other words, the probability of finding unit 3 in state /, given that unit 2 is in state P and unit 1 in state a, is the same as the probability of finding unit 3 in / given that unit 2 is in /3. The information about unit 1 does not affect the conditional probability of the state of unit 3. 

This result is a fundamental property of a Markov chain. It can be generalized to any number of units. If we know the states of units 1, 2,.. .and we ask for the probability of occurrence of a particular state at unit A + 1, all we need to know is the state of the previous unit. We can ignore the far past and retain only the near past, i.e., 

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